特征为2的平面函数的进一步研究
Further Study of Planar Functions in Characteristic Two
摘要：
平面函数在构造DES类迭代密码、纠错码、信号集和数学领域中具有重要意义。它们最初定义在奇特征的有限域上，并由Y.周\cite{Zhou}在均匀特性。2016年，L. Qu \cite{Q}提出了一种构造$\F_{2^n}$上二次平面函数的新方法。最近，D。Bartoli和M. Timpanella \cite{Bartoli}刻画了系数$a,b$的条件，使得函数$f_{a,b}(x)=ax^{2^{2m}+1}+bx^{2^m+1} \in\F_{2^{3m}}[x]$是$\F_{2^{3m}}$上的平面函数，满足Hasse-Weil界。本文利用Hasse-Weil界的推广Lang-Weil界和\cite{Q}中引入的新方法，完整地刻画了$\F_{q^k}$上四类平面函数系数的充要条件，其中$q=2^m$具有充分大的$m$（见定理\ref{main}）.它们的第一类和最后一类分别是在$\F_{q^2}$和$\F_{q^4}$上的，而另外两类是在$\F_{q^3}$上的. $\F_{q^3}$上的一个类是\cite{Bartoli}中研究的$f_{a,b}(x)$的扩展，而我们的证明似乎简单得多.另外，虽然我们的结果是$\F_{q^2}$上的平面二项式最终是一个已知的平面单项式，但我们同时也回答了这个必要性，并部分解决了\cite{Q}中提出的二项式情形的一个公开问题.
英文摘要：
Planar functions are of great importance in the constructions of DES-like iterated ciphers, error-correcting codes, signal sets and the area of mathematics. They are defined over finite fields of odd characteristic originally and generalized by Y. Zhou \cite{Zhou} in even characteristic. In 2016, L. Qu \cite{Q} proposed a new approach to constructing quadratic planar functions over $\F_{2^n}$. Very recently, D. Bartoli and M. Timpanella \cite{Bartoli} characterized the condition on coefficients $a,b$ such that the function $f_{a,b}(x)=ax^{2^{2m}+1}+bx^{2^m+1} \in\F_{2^{3m}}[x]$ is a planar function over $\F_{2^{3m}}$ by the Hasse-Weil bound. In this paper, using the Lang-Weil bound, a generalization of the Hasse-Weil bound, and the new approach introduced in \cite{Q}, we completely characterize the necessary and sufficient conditions on coefficients of four classes of planar functions over $\F_{q^k}$, where $q=2^m$ with $m$ sufficiently large (see Theorem \ref{main}). The first and last classes of them are over $\F_{q^2}$ and $\F_{q^4}$ respectively, while the other two classes are over $\F_{q^3}$. One class over $\F_{q^3}$ is an extension of $f_{a,b}(x)$ investigated in \cite{Bartoli}, while our proofs seem to be much simpler. In addition, although the planar binomial over $\F_{q^2}$ of our results is finally a known planar monomial, we also answer the necessity at the same time and solve partially an open problem for the binomial case proposed in \cite{Q}.
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